85.75.13 problem 3

Internal problem ID [22978]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 329
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:17:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} U^{\prime \prime }+\frac {2 U^{\prime }}{r}+a U&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

Order:=6; 
ode:=diff(diff(U(r),r),r)+2/r*diff(U(r),r)+a*U(r) = 0; 
dsolve(ode,U(r),type='series',r=0);
 

\[ U = c_1 \left (1-\frac {1}{6} a \,r^{2}+\frac {1}{120} a^{2} r^{4}+\operatorname {O}\left (r^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} a \,r^{2}+\frac {1}{24} a^{2} r^{4}+\operatorname {O}\left (r^{6}\right )\right )}{r} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 50

ode=D[U[r],{r,2}]+2/r*D[U[r],r]+a*U[r]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},U[r],{r,0,5}]
 

\[ U(r)\to c_1 \left (\frac {a^2 r^3}{24}-\frac {a r}{2}+\frac {1}{r}\right )+c_2 \left (\frac {a^2 r^4}{120}-\frac {a r^2}{6}+1\right ) \]

✗ Sympy

from sympy import * 
r = symbols("r") 
a = symbols("a") 
u = Function("u") 
ode = Eq(a*U(r) + Derivative(U(r), (r, 2)) + 2*Derivative(U(r), r)/r,0) 
ics = {} 
dsolve(ode,func=u(r),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE a*U(r) + Derivative(U(r), (r, 2)) + 2*Derivative(U(r), r)/r does not match hint 2nd